1959]
HOMOMORPHICIMAGES OF REAL CLANS WITH ZERO
109
3. P. S. Mostert and A. L. Shields, On the structure of semigroups on a compact
manifoldwith boundary,Ann. of Math. vol. 65 (1957)pp. 117-143.
4. G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloquium Publications,
vol. 28, 1942, New York.
Louisiana State University
ON GREEN'S THEOREM
PAUL J. COHEN
Green's theorem in two dimensons says that if C is a simple closed
curve bounding the region Q, if A(x, y) and B(c, y) are continuous
functions having derivatives, then under suitable further conditions
we have,
(1)
c
r r (dB
| Adx + Bdy = j I (-)
Jc
J Jq\
dA\
dx
dy /
dxdy
where the line integral is taken in a positive sense around the curve
C. In [l] Bochner investigated
under which conditions
(1) holds.
There it was shown that if A and B have certain regularity properties
and if the integrand on the right of (1) behaves well, then (1) does
hold. Here we shall prove (1) under what may be regarded as the
weakest possible hypotheses. This question was also treated by Shapiro in [3], and though he assumes certain regularity of A and B,
namely, the existence of the differential,
he allows certain exceptional sets which we cannot allow. The proof of our theorem is
modeled
after the proof of the Looman-Mensov
theorem
as contained,
for example, in [2]. We will not deal with the topological difficulties
involved so that our theorem will only treat the case in which Q is a
rectangle, whose sides are parallel to the coordinate axes.
Theorem.
Let A(x, y) and B(x, y) be two functions defined on the
rectangle Q, and continuous on the closure of Q. Assume further that the
partial derivatives
dA
dA
dB
dB
dx
dy
dx
dy
exist everywhere in the interior of Q, except perhaps at a countable numPresented to the Society, April 26, 1958; received by the editors March 27, 1958.
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110
P. J. COHEN
ber of points. If dB/dx—dA/dy
[February
is Lebesgue integrable in the rectangle
Q, then (1) holds.
Proof. We first need a lemma which is contained in [2]. In the
following, the word "rectangle"
means the direct product of two
intervals. Here we use the notation \S\ to represent the measure of
the set S.
Lemma. Let w(x, y) be a function defined in a square Q, such that
dw/dx and dw/dy exist at all but an enumerable set in Q. Let F be a
closed, nonempty set in Q, and N a finite constant such that
w(x, y + k) — w(x, y)\ ^ N \ k ,
ii
(2)
I w(x + h, y) — w(x, y)\ ^ N ■\ h \
whenever (x, y) belong to F and (x + h, y) and (x, y+k) belong to Q.
Let J be the smallest rectangle containing F and assume J is the product
of (ax, bx) and (a2, b2). Then
I r r
dw
rHr
I I -dxdy
- I
\ J J F dX
J a,
,
.
.
[w(x, b2) - w(x, a2)]dx g, 5N-\Q
- F\.
[w(bx,y) - w(ax,y)]dy = 57Y■
| Q- F | ,
(3)
\\
—dxdy-
j
I J J f dy
J ai
It is clearly enough to prove the theorem for all rectangles
Q'
properly contained in Q, lor, in this case, we can approximate
Q from
the interior by a sequence of such rectangles, and for each of which
(1) holds. Now by the Lebesgue integrability
of dB/dx—dA/dy
the
right side approaches
the corresponding
integral over Q, and by the
continuity
of A and B so does the left side. Hence we may assume
in the original statement
of the theorem that A and B are actually
defined in a neighborhood
of Q where they are continuous and have
derivatives at all but a countable number of points. Q will now denote
the closed rectangle. Now, let E be the set of all points P in Q, such
that (1) holds for integrations
taken over all rectangles in a suffi-
ciently small neighborhood of F. We shall show that E is all of Q.
Let F=Q—E.
Since E is obviously open, F is closed. Let Hn be the
set of all points (x, y) in Q, such that
A(x + h, y) - A(x, y)
A(x, y + k) - A(x,y)
~T~
k
B(x + h,y) - B(x, y)
B(x, y + k) - B(x, y)
h
k
~ '
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'
i959l
ON GREEN'S THEOREM
111
are all bounded by n whenever \h\ gl/»,
\k\ gl/w,
and all the
quantities involved are defined. Clearly Q, with a countable number
of exceptions, is the union of all these closed sets Hn. We may assume
that P has points in the interior of Q, since otherwise we may approximate Q by a sequence of interior squares, as described above.
Therefore by the Baire category theorem, since P is also closed, either
P contains an isolated point in the interior of Q, or there is some
square I of edge smaller than l/N, in the interior of Q, such that
IC\F is nonempty and is contained in Hn for some TV. If a rectangle
lies completely in E, then the Heine-Borel theorem shows that (1)
holds for it. Hence we see that isolated points of P cannot occur and
so the second alternative
holds. Then the conditions of the lemma
hold and we have
(4)
\ Adx+Bdy-
I Jdj
\f
(-\dxdy
J J jHf \dx
g 10A7-| / - F \.
dy }
Here J is the smallest rectangle containing IC\F and dJ denotes the
boundary of J. The set I —J is a finite union of rectangles each of
which can be approximated
from the interior by rectangles wholly
contained in P. Hence (1) holds for the set I—J, where the line
integral is taken around its boundary in the positive sense. Thus we
have
(5)
f
Jaa-j)
Adx+ Bdy= f f (-)
J Ji-j\
dx
dy /
dxdy.
So by (4) we have that
\ r
| Adx+Bdy-
(6)
I J ai
rr
If
(dB
dA\
(-)dxdy
J J (jriF)Ua-J) \dx
dy J
i
i
g ION- \l - F\ .
From (6) it follows that
/i Adx + Bdy g ION- \l\ i + jCC\dB
f-dxdy.
ai
J J11 dx
dA
dy
Now (7) holds equally well for any square I' contained in I. Thus the
set function which assigns to every square I' the quantity
fai'Adx + Bdy
is dominated by an absolutely continuous measure and hence extends to an absolutely continuous measure defined on all Borel sets.
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112
P. J. COHEN
Thus, it is given by the indefinite integral of some function. If we
can then show that the derivative of this measure in the sense of
averages taken over smaller and smaller squares is equal to dB/dx
—dA/dy almost everywhere, then we will know that (1) holds for all
rectangles in / and hence that I(~\F is empty, which is a contradiction.
Now the derivative of the measure at almost all points not in F is
clearly dB/dx—dA/dy.
This is merely the theorem concerning differentiation of indefinite integrals, since at such points (1) does hold.
On the other hand, if P is a point of density of F, then for sufficiently small squares / around P, we have that | / —F| /| /| is arbi-
trarily small. Thus from (6) it follows that (l/\l\)fdIAdx+Bdy
approaches (1/| /| )ffijnr)Uii-j)(dB/dx—dA/dy)dxdy.
This last quantity approaches the derivative of the integral taken with respect to
the sets (jr\F)KJ(I
—J). These are a regular sequence of sets in the
sense of [2 p. 106], since | (Jr\F)<U(I-J)\/\l\
tends to 1, and so
this derivative is equal to dB/dx—dA/dy
at almost all points of F.
Thus the measure is the desired indefinite integral and so (1) holds
at all points P. Now by the Heine-Borel theorem it follows that (1)
holds for the rectangle Q itself.
We may easily generalize
this result to any number of dimensions.
References
1. S. Bochner, Green-Goursat theorem, Math. Z. vol. 63 (1955) pp. 230-242.
2. S. Saks, Theory of the integral, Warsaw, 1937.
3. V. L. Shapiro, On Green's theorem, J. London Math. Soc. vol. 32 (1957) pp.
261-269.
University
of Rochester
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